module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 531,
"column": 2
} | {
"line": 532,
"column": 22
} | {
"line": 534,
"column": 0
} | [
{
"pp": "case neg.inr\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\na b p : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nha : ¬a = 0\nhb :... | [] | · apply Or.intro_left
rw [hb0, add_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 585,
"column": 6
} | {
"line": 585,
"column": 10
} | {
"line": 586,
"column": 6
} | [
{
"pp": "case e'_4\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\na✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\nq : As... | [
"case e'_4\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh : a ∣ p ^ n\na✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\nq : Associates α\n... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 599,
"column": 4
} | {
"line": 599,
"column": 8
} | {
"line": 600,
"column": 4
} | [
{
"pp": "case a\nα : Type u_1\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : UniqueFactorizationMonoid α\ninst✝² : DecidableEq (Associates α)\ninst✝¹ : (p : Associates α) → Decidable (Irreducible p)\na p : Associates α\nhp : Irreducible p\ninst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)\nn : ℕ\nh : a ∣ p ^ n\n⊢ p ^ p.count a... | [
"case a\nα : Type u_1\ninst✝⁴ : CommMonoidWithZero α\ninst✝³ : UniqueFactorizationMonoid α\ninst✝² : DecidableEq (Associates α)\ninst✝¹ : (p : Associates α) → Decidable (Irreducible p)\na p : Associates α\nhp : Irreducible p\ninst✝ : (n : ℕ) → Decidable (a ∣ p ^ n)\nn : ℕ\nh : a ∣ p ^ n\n⊢ a = p ^ p.count a.factors... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 326,
"column": 11
} | {
"line": 326,
"column": 26
} | {
"line": 326,
"column": 27
} | [
{
"pp": "R : Type u\nS : Type u_1\nA : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : Algebra R S\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : S\nh : Function.Injective ⇑(algebraMap S A)\n⊢ Transcendental R ((algebraMap S A) a) ↔ Transcendental R a",... | [
"R : Type u\nS : Type u_1\nA : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\ninst✝² : Algebra R S\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\na : S\nh : Function.Injective ⇑(algebraMap S A)\n⊢ ¬IsAlgebraic R ((algebraMap S A) a) ↔ ¬IsAlgebraic R a"
] | Transcendental, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Colimit.Module | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 48
} | {
"line": 116,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\ninst✝⁵ : Preorder ι\nG : ι → Type u_3\ninst✝⁴ : (i : ι) → AddCommMonoid (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝² : DecidableEq ι\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz w : DirectLimit G f\ni : ι\... | [] | rw [of_f, hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Colimit.Module | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 48
} | {
"line": 116,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\ninst✝⁵ : Preorder ι\nG : ι → Type u_3\ninst✝⁴ : (i : ι) → AddCommMonoid (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝² : DecidableEq ι\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz w : DirectLimit G f\ni : ι\... | [] | rw [of_f, hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Colimit.Module | {
"line": 116,
"column": 35
} | {
"line": 116,
"column": 48
} | {
"line": 116,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\ninst✝⁵ : Preorder ι\nG : ι → Type u_3\ninst✝⁴ : (i : ι) → AddCommMonoid (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝² : DecidableEq ι\ninst✝¹ : Nonempty ι\ninst✝ : IsDirectedOrder ι\nz w : DirectLimit G f\ni : ι\... | [] | rw [of_f, hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 204,
"column": 4
} | {
"line": 205,
"column": 73
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nQ : Type u_5\ninst✝¹ : AddCommGroup Q\ninst✝ : Module R Q\... | [] | rw [LinearMap.range_le_iff_comap, ← LinearMap.ker_comp,
← lTensor_comp, hfg.linearMap_comp_eq_zero, lTensor_zero, ker_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 204,
"column": 4
} | {
"line": 205,
"column": 73
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nQ : Type u_5\ninst✝¹ : AddCommGroup Q\ninst✝ : Module R Q\... | [] | rw [LinearMap.range_le_iff_comap, ← LinearMap.ker_comp,
← lTensor_comp, hfg.linearMap_comp_eq_zero, lTensor_zero, ker_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 204,
"column": 4
} | {
"line": 205,
"column": 73
} | {
"line": 207,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nQ : Type u_5\ninst✝¹ : AddCommGroup Q\ninst✝ : Module R Q\... | [] | rw [LinearMap.range_le_iff_comap, ← LinearMap.ker_comp,
← lTensor_comp, hfg.linearMap_comp_eq_zero, lTensor_zero, ker_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 487,
"column": 10
} | {
"line": 487,
"column": 32
} | {
"line": 488,
"column": 10
} | [
{
"pp": "case a.refine_4.tmul.add\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\nx✝ : A ⊗[R] B\nhx✝ : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))\na : A... | [
"case a.refine_4.tmul.add\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeLeft '' ↑I)))\na : A\nb : B\nx y ... | obtain ⟨x', hx'⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 508,
"column": 10
} | {
"line": 508,
"column": 48
} | {
"line": 509,
"column": 10
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\na : ↥(Submodule.restrictScalars R I)\nb : B\nthis : ↑a ⊗ₜ[R] b = 1 ⊗ₜ[R] b * ↑a ⊗ₜ[R] 1\n⊢ ↑a ⊗ₜ[R] 1 ∈ map includeLeft I",
"ppTerm": ... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal A\na : ↥(Submodule.restrictScalars R I)\nb : B\nthis : ↑a ⊗ₜ[R] b = 1 ⊗ₜ[R] b * ↑a ⊗ₜ[R] 1\n⊢ ↑a ∈ I"
] | apply Ideal.mem_map_of_mem includeLeft | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Flat.Basic | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 73
} | {
"line": 95,
"column": 2
} | [
{
"pp": "R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : N →ₗ[R] P\nh :\n ∀ (N' : Submodule R N) (P' : Submodule R P),\n N'.FG → P'.FG ... | [
"R : Type u\nM : Type v\nN : Type u_1\nP : Type u_2\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid P\ninst✝ : Module R P\nf : N →ₗ[R] P\nh :\n ∀ (N' : Submodule R N) (P' : Submodule R P),\n N'.FG → P'.FG → ∀ (h : N' ... | have ⟨P', Pfg, le, eq⟩ := (Nfg.map _).exists_rTensor_fg_inclusion_eq eq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 551,
"column": 10
} | {
"line": 551,
"column": 32
} | {
"line": 552,
"column": 10
} | [
{
"pp": "case a.refine_4.tmul.add\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nx✝ : A ⊗[R] B\nhx✝ : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)))\na : ... | [
"case a.refine_4.tmul.add\nR : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_2\nB : Type u_3\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nI : Ideal B\nx✝ : A ⊗[R] B\nhx : x✝ ∈ ↑(Submodule.restrictScalars R (Submodule.span (A ⊗[R] B) (⇑includeRight '' ↑I)))\na : A\nb : B\nx y... | obtain ⟨x', hx'⟩ := hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Flat.Basic | {
"line": 149,
"column": 83
} | {
"line": 150,
"column": 93
} | {
"line": 152,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Small.{v', u} R\n⊢ Flat R M ↔\n ∀ ⦃N N' : Type v'⦄ [inst : AddCommMonoid N] [inst_1 : AddCommMonoid N'] [inst_2 : Module R N] [inst_3 : Module R N']\n (f : N →ₗ[R] N'), Injective ⇑f → Injecti... | [] | by
simp_rw [iff_rTensor_preserves_injective_linearMapₛ, LinearMap.lTensor_inj_iff_rTensor_inj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.BigOperators.Expect | {
"line": 163,
"column": 47
} | {
"line": 163,
"column": 74
} | {
"line": 163,
"column": 74
} | [
{
"pp": "ι : Type u_1\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\ns : Finset ι\nf : ι → M\ni : ι\nhi : i ∈ s\nh : ∀ j ∈ s, j ≠ i → f j = 0\n⊢ (↑(#s))⁻¹ • ∑ i ∈ s, f i = (↑(#s))⁻¹ • f i",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemirin... | [
"ι : Type u_1\nM : Type u_4\ninst✝¹ : AddCommMonoid M\ninst✝ : Module ℚ≥0 M\ns : Finset ι\nf : ι → M\ni : ι\nhi : i ∈ s\nh : ∀ j ∈ s, j ≠ i → f j = 0\n⊢ (↑(#s))⁻¹ • f i = (↑(#s))⁻¹ • f i"
] | sum_eq_single_of_mem _ hi h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.Group.Finset.Gaps | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 6
} | {
"line": 44,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : CommGroup β\nF : Finset (α × α)\nk : ℕ\nh : #F = k\na b : α\nf : α → α → β\np : Fin (k + 1) → α × α := F.intervalGapsWithin h a b\n⊢ ∏ z ∈ F, f z.1 z.2 = ∏ i ∈ range k, f (p ↑i).2 (p ↑i.succ).1",
"ppTerm": "?m.40",
"assigned": true,
... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : CommGroup β\nF : Finset (α × α)\nk : ℕ\nh : #F = k\na b : α\nf : α → α → β\np : Fin (k + 1) → α × α := ⋯\n⊢ ∏ i ∈ range k, f (p ↑i).2 (p ↑i.succ).1 = ∏ z ∈ F, f z.1 z.2"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Group.EvenFunction | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 38
} | {
"line": 78,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Neg α\ninst✝ : Add β\nf g : α → β\nhf : Function.Even f\nhg : Function.Even g\n⊢ Function.Even (f + g)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"congrArg",
"instHAdd",
"Pi.instAdd",
"HAdd.hAdd",
"congr",
... | [] | intro a
simp only [hf a, hg a, Pi.add_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.EvenFunction | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 38
} | {
"line": 78,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Neg α\ninst✝ : Add β\nf g : α → β\nhf : Function.Even f\nhg : Function.Even g\n⊢ Function.Even (f + g)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"congrArg",
"instHAdd",
"Pi.instAdd",
"HAdd.hAdd",
"congr",
... | [] | intro a
simp only [hf a, hg a, Pi.add_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Group.Finset.Interval | {
"line": 42,
"column": 8
} | {
"line": 42,
"column": 21
} | {
"line": 42,
"column": 21
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommGroup R\nf : ℤ → R\n⊢ ∏ m ∈ Icc (-(↑0 + 1)) (↑0 + 1), f m = f (↑0 + 1) * f (-(↑0 + 1)) * ∏ m ∈ Icc (-↑0) ↑0, f m",
"ppTerm": "?zero",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.Icc_succ_succ",
"HMul.hMul",
"Finset.inst... | [
"case zero\nR : Type u_1\ninst✝ : CommGroup R\nf : ℤ → R\n⊢ ∏ m ∈ Icc (-↑0) ↑0 ∪ {-(↑0 + 1), ↑0 + 1}, f m = f (↑0 + 1) * f (-(↑0 + 1)) * ∏ m ∈ Icc (-↑0) ↑0, f m"
] | Icc_succ_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 32
} | {
"line": 39,
"column": 0
} | [
{
"pp": "n : ℕ\nl : List ℕ\nh : ∀ x ∈ l, x ≡ 1 [MOD n]\n⊢ l.prod ≡ 1 [MOD n]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"List.map",
"Nat.ModEq.listProd_map_one",
"Eq.mp",
"id",
"instMulNat",
"instOfNatNat",
... | [] | simpa using listProd_map_one h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 32
} | {
"line": 39,
"column": 0
} | [
{
"pp": "n : ℕ\nl : List ℕ\nh : ∀ x ∈ l, x ≡ 1 [MOD n]\n⊢ l.prod ≡ 1 [MOD n]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"List.map",
"Nat.ModEq.listProd_map_one",
"Eq.mp",
"id",
"instMulNat",
"instOfNatNat",
... | [] | simpa using listProd_map_one h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 32
} | {
"line": 39,
"column": 0
} | [
{
"pp": "n : ℕ\nl : List ℕ\nh : ∀ x ∈ l, x ≡ 1 [MOD n]\n⊢ l.prod ≡ 1 [MOD n]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Nat.instOne",
"congrArg",
"List.map",
"Nat.ModEq.listProd_map_one",
"Eq.mp",
"id",
"instMulNat",
"instOfNatNat",
... | [] | simpa using listProd_map_one h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 32
} | {
"line": 132,
"column": 0
} | [
{
"pp": "n : ℤ\nl : List ℤ\nh : ∀ x ∈ l, x ≡ 1 [ZMOD n]\n⊢ l.prod ≡ 1 [ZMOD n]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"congrArg",
"List.map",
"AddGroupWithOne.toAddMonoidWithOne",
"Eq.mp",
"id",
"Int",
"Int.ModEq.listProd_map_one",
"... | [] | simpa using listProd_map_one h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 32
} | {
"line": 132,
"column": 0
} | [
{
"pp": "n : ℤ\nl : List ℤ\nh : ∀ x ∈ l, x ≡ 1 [ZMOD n]\n⊢ l.prod ≡ 1 [ZMOD n]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"congrArg",
"List.map",
"AddGroupWithOne.toAddMonoidWithOne",
"Eq.mp",
"id",
"Int",
"Int.ModEq.listProd_map_one",
"... | [] | simpa using listProd_map_one h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 32
} | {
"line": 132,
"column": 0
} | [
{
"pp": "n : ℤ\nl : List ℤ\nh : ∀ x ∈ l, x ≡ 1 [ZMOD n]\n⊢ l.prod ≡ 1 [ZMOD n]",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"congrArg",
"List.map",
"AddGroupWithOne.toAddMonoidWithOne",
"Eq.mp",
"id",
"Int",
"Int.ModEq.listProd_map_one",
"... | [] | simpa using listProd_map_one h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Sym | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 24
} | {
"line": 187,
"column": 4
} | [
{
"pp": "case succ.refine_1\nα : Type u_1\ns : Finset α\ninst✝ : DecidableEq α\nn✝ n : ℕ\nih : ∀ {m : Sym α n}, m ∈ s.sym n ↔ ∀ a ∈ m, a ∈ s\nm : Sym α (n + 1)\na : α\nha : a ∈ s\nhe : m ∈ image (Sym.cons a) (s.sym n)\nb : α\nhb : b ∈ m\n⊢ b ∈ s",
"ppTerm": "?succ.refine_1",
"assigned": true,
"usedC... | [
"case succ.refine_1\nα : Type u_1\ns : Finset α\ninst✝ : DecidableEq α\nn✝ n : ℕ\nih : ∀ {m : Sym α n}, m ∈ s.sym n ↔ ∀ a ∈ m, a ∈ s\nm : Sym α (n + 1)\na : α\nha : a ∈ s\nhe : ∃ a_1 ∈ s.sym n, a ::ₛ a_1 = m\nb : α\nhb : b ∈ m\n⊢ b ∈ s"
] | rw [mem_image] at he | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.IsTensorProduct | {
"line": 863,
"column": 2
} | {
"line": 864,
"column": 100
} | {
"line": 866,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nA : Type u_8\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nC : Type u_11\ninst✝⁵ : CommRing C\ninst✝⁴ : Algebra R C\ninst✝³ : Algebra A C\ninst✝² : IsScalarTower R A C\nS : Type u_12\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ (↑(cancelBaseChangeAlg R S A (S ⊗[R] A... | [] | ext
simp [← TensorProduct.one_def, ← TensorProduct.tmul_one_eq_one_tmul, RingHom.algebraMap_toAlgebra] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IsTensorProduct | {
"line": 863,
"column": 2
} | {
"line": 864,
"column": 100
} | {
"line": 866,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nA : Type u_8\ninst✝⁷ : CommRing A\ninst✝⁶ : Algebra R A\nC : Type u_11\ninst✝⁵ : CommRing C\ninst✝⁴ : Algebra R C\ninst✝³ : Algebra A C\ninst✝² : IsScalarTower R A C\nS : Type u_12\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\n⊢ (↑(cancelBaseChangeAlg R S A (S ⊗[R] A... | [] | ext
simp [← TensorProduct.one_def, ← TensorProduct.tmul_one_eq_one_tmul, RingHom.algebraMap_toAlgebra] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.ReflectsIso.Basic | {
"line": 72,
"column": 18
} | {
"line": 75,
"column": 18
} | {
"line": 77,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝² : Category.{v_2, u_2} D\nE : Type u_3\ninst✝¹ : Category.{v_3, u_3} E\nF : C ⥤ D\nG : D ⥤ E\ninst✝ : (F ⋙ G).ReflectsIsomorphisms\nA✝ B✝ : C\nf : A✝ ⟶ B✝\nx✝ : IsIso (F.map f)\n⊢ IsIso f",
"ppTerm": "?m.26",
"assigned": true,
... | [] | by
rw [← isIso_iff_of_reflects_iso _ (F ⋙ G)]
dsimp
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.EqToHom | {
"line": 45,
"column": 2
} | {
"line": 46,
"column": 11
} | {
"line": 48,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : CategoryStruct.{v₁, u₁} C\nX Y : C\np : X = Y\n⊢ X ⟶ Y",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id",
... | [] | rw [p]
exact 𝟙 _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.EqToHom | {
"line": 45,
"column": 2
} | {
"line": 46,
"column": 11
} | {
"line": 48,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝ : CategoryStruct.{v₁, u₁} C\nX Y : C\np : X = Y\n⊢ X ⟶ Y",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"CategoryTheory.CategoryStruct.id",
"id",
... | [] | rw [p]
exact 𝟙 _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Whiskering | {
"line": 283,
"column": 2
} | {
"line": 283,
"column": 6
} | {
"line": 284,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nE : Type u₃\ninst✝¹ : Category.{v₃, u₃} E\nG H : C ⥤ D\nα : G ⟶ H\nF : D ⥤ E\ninst✝ : IsIso α\n⊢ inv (whiskerRight α F) = whiskerRight (inv α) F",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nE : Type u₃\ninst✝¹ : Category.{v₃, u₃} E\nG H : C ⥤ D\nα : G ⟶ H\nF : D ⥤ E\ninst✝ : IsIso α\n⊢ whiskerRight (inv α) F = inv (whiskerRight α F)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Whiskering | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 6
} | {
"line": 291,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nE : Type u₃\ninst✝¹ : Category.{v₃, u₃} E\nF : C ⥤ D\nG H : D ⥤ E\nα : G ⟶ H\ninst✝ : IsIso α\n⊢ inv (F.whiskerLeft α) = F.whiskerLeft (inv α)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [
"C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nE : Type u₃\ninst✝¹ : Category.{v₃, u₃} E\nF : C ⥤ D\nG H : D ⥤ E\nα : G ⟶ H\ninst✝ : IsIso α\n⊢ F.whiskerLeft (inv α) = inv (F.whiskerLeft α)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Equivalence | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 23
} | {
"line": 412,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\ne : C ≌ D\nF : C ⥤ E\nX : C\n⊢ (e.funInvIdAssoc F).hom.app X = F.map (e.unitInv.app X)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"CategoryThe... | [
"C : Type u₁\ninst✝² : Category.{v₁, u₁} C\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nE : Type u₃\ninst✝ : Category.{v₃, u₃} E\ne : C ≌ D\nF : C ⥤ E\nX : C\n⊢ 𝟙 (F.obj (e.inverse.obj (e.functor.obj X))) ≫ F.map (e.unitIso.inv.app X) ≫ 𝟙 (F.obj X) = F.map (e.unitInv.app X)"
] | dsimp [funInvIdAssoc] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.CategoryTheory.MorphismProperty.Basic | {
"line": 540,
"column": 43
} | {
"line": 542,
"column": 8
} | {
"line": 543,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : MorphismProperty C\nhP : ∀ (f g : Arrow C) (x : f ≅ g), P f.hom → P g.hom\nX Y Z : C\ne : X ⟶ Y\nhe : IsIso e\nf : Y ⟶ Z\nhf : P f\n⊢ P (e ≫ f)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver"... | [] | by
refine hP (Arrow.mk f) (Arrow.mk (e ≫ f)) (Arrow.isoMk (asIso (inv e)) (Iso.refl _) ?_) hf
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Endomorphism | {
"line": 149,
"column": 18
} | {
"line": 149,
"column": 42
} | {
"line": 151,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : C\nf g : (End X)ˣ\n⊢ { hom := ↑(f * g), inv := (f * g).inv, hom_inv_id := ⋯, inv_hom_id := ⋯ } =\n { hom := ↑f, inv := f.inv, hom_inv_id := ⋯, inv_hom_id := ⋯ } *\n { hom := ↑g, inv := g.inv, hom_inv_id := ⋯, inv_hom_id := ⋯ }",
"ppTerm": "?m.56",
... | [] | by cases f; cases g; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Comma.Basic | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 61
} | {
"line": 454,
"column": 62
} | [
{
"pp": "A : Type u₁\ninst✝⁶ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝⁵ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nA' : Type u₄\ninst✝³ : Category.{v₄, u₄} A'\nB' : Type u₅\ninst✝² : Category.{v₅, u₅} B'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL✝ : A ⥤ T\nR✝ : B ⥤ T\nC : Type u₄\nin... | [] | simp only [Functor.comp_map, ← F.map_comp, f.w] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Comma.Basic | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 61
} | {
"line": 454,
"column": 62
} | [
{
"pp": "A : Type u₁\ninst✝⁶ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝⁵ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nA' : Type u₄\ninst✝³ : Category.{v₄, u₄} A'\nB' : Type u₅\ninst✝² : Category.{v₅, u₅} B'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL✝ : A ⥤ T\nR✝ : B ⥤ T\nC : Type u₄\nin... | [] | simp only [Functor.comp_map, ← F.map_comp, f.w] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Comma.Basic | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 61
} | {
"line": 454,
"column": 62
} | [
{
"pp": "A : Type u₁\ninst✝⁶ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝⁵ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nA' : Type u₄\ninst✝³ : Category.{v₄, u₄} A'\nB' : Type u₅\ninst✝² : Category.{v₅, u₅} B'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL✝ : A ⥤ T\nR✝ : B ⥤ T\nC : Type u₄\nin... | [] | simp only [Functor.comp_map, ← F.map_comp, f.w] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.IsLimit | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 8
} | {
"line": 232,
"column": 4
} | [
{
"pp": "J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nr t : Cone F\nP : IsLimit r\ni : IsIso (P.lift t)\nthis✝ : IsIso (P.liftConeMorphism t).hom\nthis : IsIso (P.liftConeMorphism t)\n⊢ r ≅ t",
"ppTerm": "?m.77",
... | [
"J : Type u₁\ninst✝² : Category.{v₁, u₁} J\nK : Type u₂\ninst✝¹ : Category.{v₂, u₂} K\nC : Type u₃\ninst✝ : Category.{v₃, u₃} C\nF : J ⥤ C\nr t : Cone F\nP : IsLimit r\ni : IsIso (P.lift t)\nthis✝ : IsIso (P.liftConeMorphism t).hom\nthis : IsIso (P.liftConeMorphism t)\n⊢ t ≅ r"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Adjunction.Basic | {
"line": 703,
"column": 6
} | {
"line": 703,
"column": 69
} | {
"line": 704,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nZ : D\n⊢ (Function.Bijective fun g ↦ F.map f ≫ g) ↔ Function.Bijective fun g ↦ f ≫ g",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"E... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : C\nf : X ⟶ Y\nZ : D\n⊢ Function.Bijective (⇑(adj.homEquiv X Z) ∘ fun g ↦ F.map f ≫ g) ↔ Function.Bijective fun g ↦ f ≫ g"
] | ← Function.Bijective.of_comp_iff' (adj.homEquiv _ _).bijective, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Adjunction.Basic | {
"line": 712,
"column": 6
} | {
"line": 712,
"column": 69
} | {
"line": 713,
"column": 4
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : D\ng : X ⟶ Y\nZ : C\n⊢ (Function.Bijective fun f ↦ f ≫ G.map g) ↔ Function.Bijective fun f ↦ f ≫ g",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"E... | [
"C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX Y : D\ng : X ⟶ Y\nZ : C\n⊢ (Function.Bijective fun f ↦ f ≫ G.map g) ↔ Function.Bijective (⇑(adj.homEquiv Z Y) ∘ fun f ↦ f ≫ g)"
] | ← Function.Bijective.of_comp_iff' (adj.homEquiv _ _).bijective, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Category.Cat | {
"line": 272,
"column": 63
} | {
"line": 272,
"column": 76
} | {
"line": 274,
"column": 0
} | [
{
"pp": "C D : Cat\nF G : C ⟶ D\nh : F = G\n⊢ (eqToHom h).toNatTrans = eqToHom ⋯",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HEq.refl",
"CategoryTheory.eqT... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Cat | {
"line": 272,
"column": 63
} | {
"line": 272,
"column": 76
} | {
"line": 274,
"column": 0
} | [
{
"pp": "C D : Cat\nF G : C ⟶ D\nh : F = G\n⊢ (eqToHom h).toNatTrans = eqToHom ⋯",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"HEq.refl",
"CategoryTheory.eqT... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.LiftingProperties.Basic | {
"line": 130,
"column": 14
} | {
"line": 130,
"column": 27
} | {
"line": 130,
"column": 28
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX Y Z W X' Y' : C\nf : X ⟶ Y\nf' : X' ⟶ Y'\nh : RetractArrow f' f\ng : Z ⟶ W\ninst✝ : HasLiftingProperty g f\nu : Z ⟶ X'\nv : W ⟶ Y'\nsq : CommSq u g f' v\n⊢ (u ≫ Arrow.Hom.left h.i) ≫ f = g ≫ v ≫ Arrow.Hom.right h.i",
"ppTerm": "?m.86",
"assigned":... | [
"C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nX Y Z W X' Y' : C\nf : X ⟶ Y\nf' : X' ⟶ Y'\nh : RetractArrow f' f\ng : Z ⟶ W\ninst✝ : HasLiftingProperty g f\nu : Z ⟶ X'\nv : W ⟶ Y'\nsq : CommSq u g f' v\n⊢ (u ≫ Arrow.Hom.left h.i) ≫ f = u ≫ f' ≫ Arrow.Hom.right h.i"
] | ← sq.w_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Retract | {
"line": 169,
"column": 16
} | {
"line": 169,
"column": 42
} | {
"line": 171,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ W✝ : C\nf✝ : X✝ ⟶ Y✝\ng✝ : Z✝ ⟶ W✝\nh✝ : RetractArrow f✝ g✝\nX Y Z W : Cᵒᵖ\nf : X ⟶ Y\ng : Z ⟶ W\nh : RetractArrow f g\n⊢ Arrow.homMk (Arrow.Hom.right h.r).unop (Arrow.Hom.left h.r).unop ⋯ ≫\n Arrow.homMk (A... | [] | ext <;> simp [← unop_comp] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Retract | {
"line": 169,
"column": 16
} | {
"line": 169,
"column": 42
} | {
"line": 171,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ W✝ : C\nf✝ : X✝ ⟶ Y✝\ng✝ : Z✝ ⟶ W✝\nh✝ : RetractArrow f✝ g✝\nX Y Z W : Cᵒᵖ\nf : X ⟶ Y\ng : Z ⟶ W\nh : RetractArrow f g\n⊢ Arrow.homMk (Arrow.Hom.right h.r).unop (Arrow.Hom.left h.r).unop ⋯ ≫\n Arrow.homMk (A... | [] | ext <;> simp [← unop_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Retract | {
"line": 169,
"column": 16
} | {
"line": 169,
"column": 42
} | {
"line": 171,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nD : Type u'\ninst✝ : Category.{v', u'} D\nX✝ Y✝ Z✝ W✝ : C\nf✝ : X✝ ⟶ Y✝\ng✝ : Z✝ ⟶ W✝\nh✝ : RetractArrow f✝ g✝\nX Y Z W : Cᵒᵖ\nf : X ⟶ Y\ng : Z ⟶ W\nh : RetractArrow f g\n⊢ Arrow.homMk (Arrow.Hom.right h.r).unop (Arrow.Hom.left h.r).unop ⋯ ≫\n Arrow.homMk (A... | [] | ext <;> simp [← unop_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Functor.EpiMono | {
"line": 111,
"column": 26
} | {
"line": 114,
"column": 18
} | {
"line": 116,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝² : Category.{v₂, u₂} D\nF G : C ⥤ D\ninst✝¹ : F.PreservesMonomorphisms\nf : G ⟶ F\ninst✝ : ∀ (X : C), Mono (f.app X)\nX Y : C\nπ : X ⟶ Y\nhπ : Mono π\n⊢ Mono (G.map π)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [] | by
suffices Mono (G.map π ≫ f.app Y) from mono_of_mono (G.map π) (f.app Y)
rw [f.naturality π]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.LiftingProperties.Adjunction | {
"line": 129,
"column": 71
} | {
"line": 134,
"column": 18
} | {
"line": 136,
"column": 0
} | [
{
"pp": "C : Type u_1\nD : Type u_2\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : Category.{v_2, u_2} D\nG : C ⥤ D\nF : D ⥤ C\nadj : G ⊣ F\nA B : C\nX Y : D\ni : A ⟶ B\np : X ⟶ Y\n⊢ HasLiftingProperty (G.map i) p ↔ HasLiftingProperty i (F.map p)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
... | [] | by
constructor <;> intro <;> constructor <;> intro f g sq
· rw [← sq.left_adjoint_hasLift_iff adj]
infer_instance
· rw [← sq.right_adjoint_hasLift_iff adj]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 12
} | {
"line": 360,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g ≫ π arrows j =... | [] | apply h1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 12
} | {
"line": 360,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g ≫ π arrows j =... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 358,
"column": 4
} | {
"line": 358,
"column": 12
} | {
"line": 360,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\nf : X ⟶ B\nfs : (j : J) → X ⟶ objs j\nw : ∀ (j : J), fs j ≫ arrows j = f\ng : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g ≫ π arrows j =... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 12
} | {
"line": 374,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\nval✝ : J\... | [] | apply h1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 12
} | {
"line": 374,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\nval✝ : J\... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 372,
"column": 4
} | {
"line": 372,
"column": 12
} | {
"line": 374,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → objs j ⟶ B\ninst✝ : HasWidePullback B objs arrows\nX : C\ng1 g2 : X ⟶ widePullback B objs arrows\nh1 : ∀ (j : J), g1 ≫ π arrows j = g2 ≫ π arrows j\nh2 : g1 ≫ base arrows = g2 ≫ base arrows\nval✝ : J\... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 12
} | {
"line": 542,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : C\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B objs arrows ⟶ X\nh1 : ∀ (j : J), ι arrows j ≫ g = f... | [] | apply h1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 12
} | {
"line": 542,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : C\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B objs arrows ⟶ X\nh1 : ∀ (j : J), ι arrows j ≫ g = f... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 12
} | {
"line": 542,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : C\nf : B ⟶ X\nfs : (j : J) → objs j ⟶ X\nw : ∀ (j : J), arrows j ≫ fs j = f\ng : widePushout B objs arrows ⟶ X\nh1 : ∀ (j : J), ι arrows j ≫ g = f... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 12
} | {
"line": 559,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : C\ng1 g2 : widePushout B objs arrows ⟶ X\nh1 : ∀ (j : J), ι arrows j ≫ g1 = ι arrows j ≫ g2\nh2 : head arrows ≫ g1 = head arrows ≫ g2\nval✝ : J\n⊢... | [] | apply h1 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 12
} | {
"line": 559,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : C\ng1 g2 : widePushout B objs arrows ⟶ X\nh1 : ∀ (j : J), ι arrows j ≫ g1 = ι arrows j ≫ g2\nh2 : head arrows ≫ g1 = head arrows ≫ g2\nval✝ : J\n⊢... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 12
} | {
"line": 559,
"column": 0
} | [
{
"pp": "case some\nJ : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\nB : C\nobjs : J → C\narrows : (j : J) → B ⟶ objs j\ninst✝ : HasWidePushout B objs arrows\nX : C\ng1 g2 : widePushout B objs arrows ⟶ X\nh1 : ∀ (j : J), ι arrows j ≫ g1 = ι arrows j ≫ g2\nh2 : head arrows ≫ g1 = head arrows ≫ g2\nval✝ : J\n⊢... | [] | apply h1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Products | {
"line": 849,
"column": 4
} | {
"line": 849,
"column": 53
} | {
"line": 850,
"column": 4
} | [
{
"pp": "case refine_1\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ne : X ≅ Y\nJ : Type u_1\ninst✝ : Unique J\ns : Fan fun x ↦ Y\nj : J\n⊢ (s.proj default ≫ e.inv) ≫ (mk X fun x ↦ e.hom).proj j = s.proj j",
"ppTerm": "?refine_1",
"assigned": true,
"usedCons... | [
"case refine_1\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ne : X ≅ Y\nJ : Type u_1\ninst✝ : Unique J\ns : Fan fun x ↦ Y\n⊢ (s.proj default ≫ e.inv) ≫ (mk X fun x ↦ e.hom).proj default = s.proj default"
] | obtain rfl : j = default := Subsingleton.elim _ _ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Limits.Shapes.Products | {
"line": 889,
"column": 2
} | {
"line": 889,
"column": 6
} | {
"line": 890,
"column": 2
} | [
{
"pp": "β : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Unique β\nf : β → C\n⊢ Sigma.ι f default ≫ (coproductUniqueIso f).hom = eqToHom ⋯",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Inhabited.default",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Ho... | [
"β : Type w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Unique β\nf : β → C\n⊢ eqToHom ⋯ = Sigma.ι f default ≫ (coproductUniqueIso f).hom"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Products | {
"line": 899,
"column": 4
} | {
"line": 899,
"column": 53
} | {
"line": 900,
"column": 4
} | [
{
"pp": "case refine_1\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ne : X ≅ Y\nJ : Type u_1\ninst✝ : Unique J\ns : Cofan fun x ↦ X\nj : J\n⊢ (mk Y fun x ↦ e.hom).inj j ≫ e.inv ≫ s.inj default = s.inj j",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstan... | [
"case refine_1\nβ : Type w\nα : Type w₂\nγ : Type w₃\nC : Type u\ninst✝¹ : Category.{v, u} C\nX Y : C\ne : X ≅ Y\nJ : Type u_1\ninst✝ : Unique J\ns : Cofan fun x ↦ X\n⊢ (mk Y fun x ↦ e.hom).inj default ≫ e.inv ≫ s.inj default = s.inj default"
] | obtain rfl : j = default := Subsingleton.elim _ _ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Limits.Shapes.Products | {
"line": 959,
"column": 12
} | {
"line": 959,
"column": 20
} | {
"line": 961,
"column": 0
} | [
{
"pp": "β : Type w\nC : Type u\ninst✝² : Category.{v, u} C\nγ : Type w'\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ⇑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' ⋯) ≫ colimit.ι (Discrete.functor f) { as := ε b } =\n colimit.ι (Discrete.functor f) { as := ε (ε.s... | [] | { simp } | Lean.Elab.Tactic.evalTacticSeqBracketed | Lean.Parser.Tactic.tacticSeqBracketed |
Mathlib.CategoryTheory.Limits.Shapes.Products | {
"line": 959,
"column": 12
} | {
"line": 959,
"column": 20
} | {
"line": 961,
"column": 0
} | [
{
"pp": "β : Type w\nC : Type u\ninst✝² : Category.{v, u} C\nγ : Type w'\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ⇑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' ⋯) ≫ colimit.ι (Discrete.functor f) { as := ε b } =\n colimit.ι (Discrete.functor f) { as := ε (ε.s... | [] | { simp } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Products | {
"line": 959,
"column": 12
} | {
"line": 959,
"column": 20
} | {
"line": 961,
"column": 0
} | [
{
"pp": "case p\nβ : Type w\nC : Type u\ninst✝² : Category.{v, u} C\nγ : Type w'\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ⇑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' ⋯) ≫ colimit.ι (Discrete.functor f) { as := ε b } =\n colimit.ι (Discrete.functor f) { as :... | [] | { simp } | Lean.Elab.Tactic.evalTacticSeqBracketed | Lean.Parser.Tactic.tacticSeqBracketed |
Mathlib.CategoryTheory.Limits.Shapes.Products | {
"line": 959,
"column": 12
} | {
"line": 959,
"column": 20
} | {
"line": 961,
"column": 0
} | [
{
"pp": "case p\nβ : Type w\nC : Type u\ninst✝² : Category.{v, u} C\nγ : Type w'\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ⇑ε)\nb : β\nh :\n (Discrete.functor f).map (Discrete.eqToHom' ⋯) ≫ colimit.ι (Discrete.functor f) { as := ε b } =\n colimit.ι (Discrete.functor f) { as :... | [] | { simp } | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {
"line": 512,
"column": 4
} | {
"line": 513,
"column": 20
} | {
"line": 514,
"column": 4
} | [
{
"pp": "case uniq.h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nX✝ Y✝ X Y X' : C\nc : BinaryCofan X Y\nf : X' ⟶ X\ninst✝ : IsIso f\nh : IsColimit c\ns : BinaryCofan X' ((pair X Y).obj { as := right })\nm : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left } ⟶ s.pt\ne₁ : (f ≫ c.inl) ≫ m = s.inl\n... | [
"case uniq.h₂\nC : Type u\ninst✝¹ : Category.{v, u} C\nX✝ Y✝ X Y X' : C\nc : BinaryCofan X Y\nf : X' ⟶ X\ninst✝ : IsIso f\nh : IsColimit c\ns : BinaryCofan X' ((pair X Y).obj { as := right })\nm : ((Functor.const (Discrete WalkingPair)).obj c.pt).obj { as := left } ⟶ s.pt\ne₁ : (f ≫ c.inl) ≫ m = s.inl\ne₂ : c.inr ≫... | · rw [← cancel_epi f]
simpa using e₁ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {
"line": 1538,
"column": 4
} | {
"line": 1541,
"column": 24
} | {
"line": 1542,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y Z P✝ : C\nsXY : BinaryFan X Y\nsYZ : BinaryFan Y Z\nP : IsLimit sXY\nQ : IsLimit sYZ\ns : BinaryFan sXY.pt Z\nR : IsLimit s\nt : Cone (pair X sYZ.pt)\n⊢ ∀ (j : Discrete WalkingPair), R.lift (BinaryFan.assocInv P t) ≫ (BinaryFan.assoc Q s).π.app j = t.π.app j",... | [] | rintro ⟨⟨⟩⟩
· simp
apply Q.hom_ext
rintro ⟨⟨⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {
"line": 1538,
"column": 4
} | {
"line": 1541,
"column": 24
} | {
"line": 1542,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y Z P✝ : C\nsXY : BinaryFan X Y\nsYZ : BinaryFan Y Z\nP : IsLimit sXY\nQ : IsLimit sYZ\ns : BinaryFan sXY.pt Z\nR : IsLimit s\nt : Cone (pair X sYZ.pt)\n⊢ ∀ (j : Discrete WalkingPair), R.lift (BinaryFan.assocInv P t) ≫ (BinaryFan.assoc Q s).π.app j = t.π.app j",... | [] | rintro ⟨⟨⟩⟩
· simp
apply Q.hom_ext
rintro ⟨⟨⟩⟩ <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 580,
"column": 46
} | {
"line": 582,
"column": 35
} | {
"line": 584,
"column": 0
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nT : D\nS : C ⥤ D\nX Y : CostructuredArrow S T\nh : X = Y\n⊢ (eqToHom h).left = eqToHom ⋯",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiv... | [] | by
subst h
simp only [eqToHom_refl, id_left] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 33
} | {
"line": 100,
"column": 34
} | [
{
"pp": "X Y Z W : WalkingParallelPair\nf : WalkingParallelPairHom X Y\ng : WalkingParallelPairHom Y Z\nh : WalkingParallelPairHom Z W\n⊢ (f.comp g).comp h = f.comp (g.comp h)",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.WalkingParallelPair",
"Category... | [
"case left.id.id\n⊢ (left.comp (id one)).comp (id one) = left.comp ((id one).comp (id one))",
"case right.id.id\n⊢ (right.comp (id one)).comp (id one) = right.comp ((id one).comp (id one))",
"case id.left.id\n⊢ ((id zero).comp left).comp (id one) = (id zero).comp (left.comp (id one))",
"case id.right.id\n⊢ ((... | cases f <;> cases g <;> cases h | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 14
} | {
"line": 102,
"column": 8
} | [
{
"pp": "case h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ Z\ninst✝ : Mono h\nx : X ⟶ W\ny : Y ⟶ W\nhxh : x ≫ h = f\nhyh : y ≫ h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : t.fst ≫ x ≫ h = t.snd ≫ y ≫ h\nm✝ : t.pt ⟶ (m... | [
"case h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ Z\ninst✝ : Mono h\nx : X ⟶ W\ny : Y ⟶ W\nhxh : x ≫ h = f\nhyh : y ≫ h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : t.fst ≫ x ≫ h = t.snd ≫ y ≫ h\nm✝ : t.pt ⟶ (mk s.fst s.sn... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 101,
"column": 10
} | {
"line": 101,
"column": 14
} | {
"line": 102,
"column": 8
} | [
{
"pp": "case h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ Z\ninst✝ : Mono h\nx : X ⟶ W\ny : Y ⟶ W\nhxh : x ≫ h = f\nhyh : y ≫ h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : t.fst ≫ x ≫ h = t.snd ≫ y ≫ h\nm✝ : t.pt ⟶ (m... | [
"case h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Z\ng✝ : Y ⟶ Z\nf : X ⟶ Z\ng : Y ⟶ Z\nh : W ⟶ Z\ninst✝ : Mono h\nx : X ⟶ W\ny : Y ⟶ W\nhxh : x ≫ h = f\nhyh : y ≫ h = g\ns : PullbackCone f g\nhs : IsLimit s\nt : PullbackCone x y\nthis : t.fst ≫ x ≫ h = t.snd ≫ y ≫ h\nm✝ : t.pt ⟶ (mk s.fst s.sn... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Comma.Over.Basic | {
"line": 920,
"column": 15
} | {
"line": 920,
"column": 27
} | {
"line": 920,
"column": 28
} | [
{
"pp": "T : Type u₁\ninst✝² : Category.{v₁, u₁} T\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nX : T\nf✝ g✝ : Under X\nφ : f✝ ⟶ g✝\nf g : Under X\nk : f ⟶ g\ninst✝ : Epi k\nY : T\nl m : g.right ⟶ Y\na : Hom.right k ≫ l = Hom.right k ≫ m\n⊢ g.hom ≫ l = g.hom ≫ m",
"ppTerm": "?m.59",
"assigned": true,
... | [
"T : Type u₁\ninst✝² : Category.{v₁, u₁} T\nD : Type u₂\ninst✝¹ : Category.{v₂, u₂} D\nX : T\nf✝ g✝ : Under X\nφ : f✝ ⟶ g✝\nf g : Under X\nk : f ⟶ g\ninst✝ : Epi k\nY : T\nl m : g.right ⟶ Y\na : Hom.right k ≫ l = Hom.right k ≫ m\n⊢ (f.hom ≫ Hom.right k) ≫ l = (f.hom ≫ Hom.right k) ≫ m"
] | ← Under.w k, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 277,
"column": 10
} | {
"line": 277,
"column": 14
} | {
"line": 278,
"column": 10
} | [
{
"pp": "case h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\nh : X ⟶ W\ninst✝ : Epi h\nx : W ⟶ Y\ny : W ⟶ Z\nhhx : h ≫ x = f\nhhy : h ≫ y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm✝ : (mk s.inl s.inr ⋯).pt ⟶ t.pt\nhr : s.inl ≫ m... | [
"case h₀\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\nh : X ⟶ W\ninst✝ : Epi h\nx : W ⟶ Y\ny : W ⟶ Z\nhhx : h ≫ x = f\nhhy : h ≫ y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm✝ : (mk s.inl s.inr ⋯).pt ⟶ t.pt\nhr : s.inl ≫ m✝ = t.inl\nh... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono | {
"line": 281,
"column": 10
} | {
"line": 281,
"column": 14
} | {
"line": 282,
"column": 10
} | [
{
"pp": "case h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\nh : X ⟶ W\ninst✝ : Epi h\nx : W ⟶ Y\ny : W ⟶ Z\nhhx : h ≫ x = f\nhhy : h ≫ y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm✝ : (mk s.inl s.inr ⋯).pt ⟶ t.pt\nhr : s.inl ≫ m... | [
"case h₁\nC : Type u\ninst✝¹ : Category.{v, u} C\nW X Y Z : C\nf✝ : X ⟶ Y\ng✝ : X ⟶ Z\nf : X ⟶ Y\ng : X ⟶ Z\nh : X ⟶ W\ninst✝ : Epi h\nx : W ⟶ Y\ny : W ⟶ Z\nhhx : h ≫ x = f\nhhy : h ≫ y = g\ns : PushoutCocone f g\nhs : IsColimit s\nt : PushoutCocone x y\nm✝ : (mk s.inl s.inr ⋯).pt ⟶ t.pt\nhr : s.inl ≫ m✝ = t.inl\nh... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 149,
"column": 7
} | {
"line": 149,
"column": 41
} | {
"line": 149,
"column": 41
} | [
{
"pp": "⊢ ∀ {X Y : WalkingParallelPair} (f : X ⟶ Y),\n (𝟭 WalkingParallelPair).map f ≫ (eqToIso ⋯).hom =\n (eqToIso ⋯).hom ≫ (walkingParallelPairOp ⋙ walkingParallelPairOp.leftOp).map f",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Opposite",
"CategoryTheory.Catego... | [] | by rintro _ _ (_ | _ | _) <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 281,
"column": 64
} | {
"line": 281,
"column": 98
} | {
"line": 281,
"column": 98
} | [
{
"pp": "C : Type u\nX Y : C\ninst✝ : Category.{v, u} C\nF : WalkingParallelPair ⥤ C\n⊢ ∀ {X Y : WalkingParallelPair} (f : X ⟶ Y),\n F.map f ≫ (eqToIso ⋯).hom = (eqToIso ⋯).hom ≫ (parallelPair (F.map left) (F.map right)).map f",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"Categ... | [] | by rintro _ _ (_ | _ | _) <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 433,
"column": 35
} | {
"line": 433,
"column": 63
} | {
"line": 433,
"column": 63
} | [
{
"pp": "C : Type u\nX Y : C\ninst✝ : Category.{v, u} C\nf g : X ⟶ Y\nt : Cofork f g\n⊢ t.ι.app zero = g ≫ t.π",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"Category... | [
"C : Type u\nX Y : C\ninst✝ : Category.{v, u} C\nf g : X ⟶ Y\nt : Cofork f g\n⊢ t.ι.app zero = t.ι.app zero"
] | ← t.app_zero_eq_comp_π_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 656,
"column": 23
} | {
"line": 656,
"column": 57
} | {
"line": 656,
"column": 58
} | [
{
"pp": "C : Type u\nX Y : C\ninst✝ : Category.{v, u} C\nf g : X ⟶ Y\nF : WalkingParallelPair ⥤ C\nt : Cone F\n⊢ ∀ ⦃X Y : WalkingParallelPair⦄ (f : X ⟶ Y),\n ((Functor.const WalkingParallelPair).obj t.pt).map f ≫ t.π.app Y ≫ eqToHom ⋯ =\n (t.π.app X ≫ eqToHom ⋯) ≫ (parallelPair (F.map left) (F.map right... | [] | by rintro _ _ (_ | _ | _) <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 666,
"column": 23
} | {
"line": 666,
"column": 57
} | {
"line": 666,
"column": 58
} | [
{
"pp": "C : Type u\nX Y : C\ninst✝ : Category.{v, u} C\nf g : X ⟶ Y\nF : WalkingParallelPair ⥤ C\nt : Cocone F\n⊢ ∀ ⦃X Y : WalkingParallelPair⦄ (f : X ⟶ Y),\n (parallelPair (F.map left) (F.map right)).map f ≫ eqToHom ⋯ ≫ t.ι.app Y =\n (eqToHom ⋯ ≫ t.ι.app X) ≫ ((Functor.const WalkingParallelPair).obj t... | [] | by rintro _ _ (_ | _ | _) <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 454,
"column": 2
} | {
"line": 454,
"column": 6
} | {
"line": 455,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX Y : C\n⊢ 𝟙 X = 0 ∧ 𝟙 Y = 0 ≃ (X ≅ 0) × (Y ≅ 0)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver... | [
"C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX Y : C\n⊢ (X ≅ 0) × (Y ≅ 0) ≃ 𝟙 X = 0 ∧ 𝟙 Y = 0"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 22
} | {
"line": 184,
"column": 4
} | [
{
"pp": "case id\nC : Type u\ninst✝ : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\n⊢ Hom.id j ∈\n Option.casesOn j (Option.casesOn j {Hom.id none} fun j ↦ {Hom.term j}) fun j' ↦\n if h : some j' = j then ⋯.mpr {Hom.id j} else ∅",
"ppTerm": "?id",
"assigned": true,
"usedConstants":... | [] | cases j <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 22
} | {
"line": 184,
"column": 4
} | [
{
"pp": "case id\nC : Type u\ninst✝ : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\n⊢ Hom.id j ∈\n Option.casesOn j (Option.casesOn j {Hom.id none} fun j ↦ {Hom.term j}) fun j' ↦\n if h : some j' = j then ⋯.mpr {Hom.id j} else ∅",
"ppTerm": "?id",
"assigned": true,
"usedConstants":... | [] | cases j <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 22
} | {
"line": 184,
"column": 4
} | [
{
"pp": "case id\nC : Type u\ninst✝ : Category.{v, u} C\nJ : Type v\nj : WidePullbackShape J\n⊢ Hom.id j ∈\n Option.casesOn j (Option.casesOn j {Hom.id none} fun j ↦ {Hom.term j}) fun j' ↦\n if h : some j' = j then ⋯.mpr {Hom.id j} else ∅",
"ppTerm": "?id",
"assigned": true,
"usedConstants":... | [] | cases j <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 22
} | {
"line": 206,
"column": 4
} | [
{
"pp": "case id\nC : Type u\ninst✝ : Category.{v, u} C\nJ : Type v\nj : WidePushoutShape J\n⊢ Hom.id j ∈\n Option.casesOn j (Option.casesOn j {Hom.id none} fun j' ↦ {Hom.init j'}) fun j_1 ↦\n if h : some j_1 = j then ⋯.mpr {Hom.id j} else ∅",
"ppTerm": "?id",
"assigned": true,
"usedConstant... | [] | cases j <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 22
} | {
"line": 206,
"column": 4
} | [
{
"pp": "case id\nC : Type u\ninst✝ : Category.{v, u} C\nJ : Type v\nj : WidePushoutShape J\n⊢ Hom.id j ∈\n Option.casesOn j (Option.casesOn j {Hom.id none} fun j' ↦ {Hom.init j'}) fun j_1 ↦\n if h : some j_1 = j then ⋯.mpr {Hom.id j} else ∅",
"ppTerm": "?id",
"assigned": true,
"usedConstant... | [] | cases j <;> simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits | {
"line": 205,
"column": 6
} | {
"line": 205,
"column": 22
} | {
"line": 206,
"column": 4
} | [
{
"pp": "case id\nC : Type u\ninst✝ : Category.{v, u} C\nJ : Type v\nj : WidePushoutShape J\n⊢ Hom.id j ∈\n Option.casesOn j (Option.casesOn j {Hom.id none} fun j' ↦ {Hom.init j'}) fun j_1 ↦\n if h : some j_1 = j then ⋯.mpr {Hom.id j} else ∅",
"ppTerm": "?id",
"assigned": true,
"usedConstant... | [] | cases j <;> simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 15
} | {
"line": 428,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y Z : C\nf g : X ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasKernel g\nh : f = g\ne : Z ⟶ X\nhe : e ≫ g = 0\n⊢ kernel.lift g e he ≫ (kernelIsoOfEq h).inv = kernel.lift f e ⋯",
"ppTerm": "?m.68",
"assigned": true,
"usedConstant... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 15
} | {
"line": 428,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y Z : C\nf g : X ⟶ Y\ninst✝¹ : HasKernel f\ninst✝ : HasKernel g\nh : f = g\ne : Z ⟶ X\nhe : e ≫ g = 0\n⊢ kernel.lift g e he ≫ (kernelIsoOfEq h).inv = kernel.lift f e ⋯",
"ppTerm": "?m.68",
"assigned": true,
"usedConstant... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 930,
"column": 2
} | {
"line": 930,
"column": 15
} | {
"line": 932,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\n⊢ cokernel.π f ≫ (cokernelIsoOfEq h).hom = cokernel.π g",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"CategoryTheory.Catego... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 930,
"column": 2
} | {
"line": 930,
"column": 15
} | {
"line": 932,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\n⊢ cokernel.π f ≫ (cokernelIsoOfEq h).hom = cokernel.π g",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"CategoryTheory.Catego... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 935,
"column": 2
} | {
"line": 935,
"column": 15
} | {
"line": 937,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\n⊢ cokernel.π g ≫ (cokernelIsoOfEq h).inv = cokernel.π f",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"CategoryTheory.Catego... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 935,
"column": 2
} | {
"line": 935,
"column": 15
} | {
"line": 937,
"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\n⊢ cokernel.π g ≫ (cokernelIsoOfEq h).inv = cokernel.π f",
"ppTerm": "?m.57",
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"usedConstants": [
"CategoryTheory.Catego... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 941,
"column": 2
} | {
"line": 941,
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"column": 0
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y Z : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\ne : Y ⟶ Z\nhe : g ≫ e = 0\n⊢ (cokernelIsoOfEq h).hom ≫ cokernel.desc g e he = cokernel.desc f e ⋯",
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"us... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.Kernels | {
"line": 941,
"column": 2
} | {
"line": 941,
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} | {
"line": 943,
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} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasZeroMorphisms C\nX Y Z : C\nf g : X ⟶ Y\ninst✝¹ : HasCokernel f\ninst✝ : HasCokernel g\nh : f = g\ne : Y ⟶ Z\nhe : g ≫ e = 0\n⊢ (cokernelIsoOfEq h).hom ≫ cokernel.desc g e he = cokernel.desc f e ⋯",
"ppTerm": "?m.65",
"assigned": true,
"us... | [] | cases h; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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